Inverse spectral problems for energy-dependent Sturm-Liouville equations with δ-interaction

Abstract

In this study, inverse spectral problems for a energy-dependent Sturm-Liouville equations with ?-interaction on a finite interval are considered. Some useful integral representations for the solutions of the considered equation have been derived and using these, properties of the spectral characteristics of the boundary value problem are investigated. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data, and from two spectra are proved.

On the Uniqueness of Inverse Problems for ”weighted” Sturm-Liouville Operator With Delta Interaction

“…If λ n = λ n , α n = α n , n = 1, 2..., L = L. Thus the specification of the spectral data Ω = {λ n , α n } ∞ n=1 uniquely determines the operator L.Proof. Under the hypothesis of the theorem we obtain, in view of(11), thatM (λ) = M (λ) on consequently by Theorem 1, L = L. ◀ If λ n = λ n and µ n = µ n , n = 1, 2..., then L = L. Thus the specification of two spectra {λ n , µ n } ∞ n=1 uniquely determines L.Proof. It is obvious that characteristic functions ∆(λ) and ψ(0, λ) are uniquely determined by the sequences{λ n } ∞ n=1 and {µ n } ∞ n=1 , respectively.…”

“…Inverse Sturm-Liuville problems with interior point conditions depending on the spectral parameter are less to investigate, and nowadays there are only a rather limited number of papers in this direction (see [5], [8], [10]- [12], and the references therein).…”

Inverse Nodal Problems for Pencils of Singular Sturm-Liouville Operators

“…In this section, the solution of the nodal inverse problem for the diffusion operator with p(x) = βδ (x − a)-Dirac delta potential and any of the set of nodal points dense in the interval (0, π) of the constants β, h, H and q(x) function, an algorithm for determining with the help of subsequence will be given. Such problems have been studied in studies of [3], [15], [21], [22] for the regular diffusion operator.…”

Pozitif Yarı Eksende Süreksizlik Koşuluna Sahip Sturm-Liouville Operatörünün Ters Saçılma Problemi

“…Inverse problem for a wave equation with piecewise constant coefficient was worked in (Lavrent'ev Jr, 1992). In case of the Sturm-Liouville equation with discontinuity conditions (or transmission conditions) at a point on the positive half line, the direct and inverse scattering problem with various boundary conditions and discontinuity conditions were investigated in (Huseynov and Osmanova, 2007;Huseynov and Osmanli, 2009;Huseynov and Mammadova, 2013;Manafov and Kablan, 2013). Moreover, the direct and inverse scattering problem for Sturm-Liouville operator with nonlinear spectral parameter in the boundary conditions were studied in (Goktas and Mamedov, 2020;Mamedov, 2009;Mamedov and Kosar, 2010;Mamedov and Kosar, 2011).…”